Theorem breq12 4035

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 4033	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 4034	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   class class class wbr 4030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031

This theorem is referenced by:  breq12i  4039  breq12d  4043  breqan12d  4046  posng  4732  isopolem  5866  poxp  6287  rbropapd  6297  ecopover  6689  ecopoverg  6692  ltdcnq  7459  recexpr  7700  ltresr  7901  reapval  8597  ltxr  9844  xrltnr  9848  xrltnsym  9862  xrlttr  9864  xrltso  9865  xrlttri3  9866  xposdif  9951  f1olecpbl  12899  exmidsbthrlem  15582